Reflector Antennas

Paraboloidal Reflectors

Antennas useful for radio astronomy at
short wavelengths must have
collecting areas much larger than the collecting area $\lambda^2 / (4
\pi)$ of an isotropic antenna and much higher angular resolution than a
short dipole provides. Since arrays of dipoles are
impractical at wavelengths $\lambda < 1$ m or so, most radio
telescopes use large reflectors
to collect and focus
power onto the simple feed
antennas, such as waveguide horns or dipoles
backed by
small reflectors, that are connected to receivers. The most common
reflector
shape is a paraboloid of revolution because it can focus the plane
wave from a distant point source
onto a single focal
point.

The reflector shape that can focus
plane waves onto a single
point must keep all parts of an on-axis plane wavefront in phase at
the focal point.
Thus the
total path lengths to the focus must
all be the same, and this requirement is sufficient to determine the
shape
of the desired
reflecting
surface. Clearly the surface must be rotationally symmetric about its
axis. In any plane containing the axis, the surface looks like the
curve below.

A plane
containing the axis of a
paraboloidal
reflector with focal length $f$. Plane wave fronts from a
distant point source are shown as dotted lines perpendicular to the $z$
axis. From a wavefront at height $h$ above the vertex, the ray
path (dashed line) lengths at all radial offsets $r$ down to the
reflector and up to the focal point at $z = f$ must be the same.

The requirement of constant path
length can be written by equating
the on-axis path length $(f + h)$ from any height $h$ to the reflector
and then back to the focus at height $f$ with the off-axis path length:
$$(f +
h) = \sqrt{r^2 + (f - z)^2} +(h - z)~.$$ We need to extract the
reflector height $z$ as a function of radius $r$:
$$ \sqrt{r^2 + (f
-z)^2} +h - z = f + h$$ $$ r^2 + f^2 + z^2 - 2fz = f^2 + z^2 + 2fz
$$ The result is
$$\bbox[border:3px blue solid,7pt]{z = {r^2 \over 4 f}}\rlap{\quad \rm
{(3B1)}}$$
This is the equation of a paraboloid with
focal
length $f$. The ratio of the focal length $f$ to the
diameter $D$ of the reflector is
called the $f/D$
ratio or focal
ratio. In principle it
is a free parameter for the telescope designer. In practice it is
constrained. If $f/D$ is too high, the support structure needed
to hold the feed or subreflector at the focus becomes unwieldy.
Thus most large radio telescopes have $f/D \approx 0.4$, an unusually
low focal ratio by optical standards. The drawback of a low $f/D$
is a small field of view. The focal
ellipsoid is the volume around the exact focal point that
remains in reasonably good focus. Only a small number (about
seven) feeds can fit inside the focal ellipsoid of an $f/D \approx 0.4$
paraboloid. Large arrays of feeds or imaging cameras require
larger $f/D$ ratios, obtained either by flattening the paraboloid or by
using magnifying subreflectors to increase the effective focal
length.

Consequently, the primary mirrors of
most radio telescopes are circular
paraboloids or sections thereof. Their advantages are:

The
effective collecting area $A_{\rm e}$ of a reflector antenna can
approach its
projected geometric area $A = \pi D^2 / 4$.

Electrical simplicity
(compared with a phased array of dipoles, for example).

A single
reflector works over a wide range of frequencies. Changing frequencies
only requires changing the feed antenna and receiver located at the
focal
point, not building a whole new radio telescope.

The Far-field Distance

How far away must a point source
be for the received waves to
satisfy our assumption that they are nearly planar? The answer depends
on both the wavelength $\lambda$ and the reflector diameter $D$.
Consider the spherical wave emitted by a point source a finite distance
$R$
from the reflector.

The
spherical wave emitted by a point
source at distance $R$ deviates from a plane by $\Delta$ at the edge of
an aperture whose size is $D$.

Such a large far-field distance makes
ground-based measurements of
the GBT antenna pattern impractical. To measure the shape of the GBT
reflector
surface using radio holography, we can observe a geostationary
satellite having an orbital altitude $R > 2000$ km. Similarly, to
determine
the transmitting power pattern for a large radar antenna such as the $D
= 305$ m Arecibo reflector, we can passively observe a celestial point
source in the far field and use the reciprocity theorem to equate the
transmitting and receiving patterns.

Patterns of Aperture Antennas

In optics, the term aperture
refers to the opening
through which
all rays pass. For example, the aperture of a paraboloidal reflector
antenna would be the plane circle, normal to the rays from a distant
point source, that just covers the paraboloid. The phase of the plane
wave from a distant point source would be constant across the aperture
plane when the aperture is perpendicular to the line-of-sight.

The aperture
plane associated with a paraboloidal dish of diameter $D$.

Another example of an aperture is the
mouth of a
waveguide horn antenna.

"Doc" Ewen
looking into the rectangular aperture of the horn antenna used to
discover the $\lambda = 21$ cm line of neutral hydrogen.

How can we calculate the beam
pattern, or power gain as a
function of
direction, of an aperture antenna? For simplicity, we first consider a
one-dimensional aperture of width $D$ and calculate the electric field pattern
at a distant ($R \gg R_{\rm ff}$) point.

Coordinate
system for a linear aperture extending from $x = -D/2$ to $x = +D/2$.

Treating this as a transmitting
antenna, we imagine that the feed
illuminates the aperture with a sine wave of fixed frequency $\nu =
\omega /
(2 \pi)$ and electric field strength $g(x)$ that varies across the
aperture. The
illumination induces currents in the reflector. The current densities
$J$ will vary with both position and time:
$$J \propto g(x)
\exp(-i\omega t)~.$$ We don't need to worry about the constant of
proportionality yet; it can be calculated later from energy
conservation. Although we are interested only in electromagnetic
radiation, we use
Huygen's principle
that applies to waves of
any type, sound waves for example. It asserts that the aperture can be
treated as a collection of small elements which act individually as
small antennas.
The electric field produced by the whole aperture at large distances is
just the vector sum of the elemental electric fields from these small
antennas. The field from
each element extending from $x$ to $x + dx$ is:
$$df \propto g(x)
{\exp(-i2\pi r / \lambda) \over r} dx~.$$
At large distances compared
with the aperture size ($r \gg D$) we can make the
Fraunhofer
approximation
$$r \approx R
+ x \sin\theta = R + x l~,$$ where $$\bbox[border:3px blue solid,7pt]{l
\equiv \sin\theta}\rlap{\quad \rm
{(3B3)}}$$ At large
distances, the quantity
$$ {1 \over r} \approx {1 \over R}$$ is nearly constant across the
aperture and can be absorbed in the
constant of proportionality. However, the periodic term cannot be
ignored at any distance: $$df
\propto g(x) \exp(-i 2 \pi R / \lambda) \exp( -i 2 \pi x l / \lambda) d
x$$ Note that $\exp(-i 2 \pi R / \lambda)$ is a constant since $R$ is
fixed, so this factor can be absorbed by the constant of
proportionality. If we define
$$\bbox[border:3px blue solid,7pt]{u \equiv {x \over
\lambda}}\rlap{\quad \rm {(3B4)}}$$ to express
position along the aperture in
units of wavelength, then $$\bbox[border:3px blue solid,7pt]{f(l) =
\int_{\rm aperture} g(u)
e^{-i 2 \pi l u} du}\rlap{\quad \rm {(3B5)}}$$
In words, this very important equation says:

In the far field, the
electric-field pattern of
an aperture antenna is
the Fourier transform of the electric field illuminating the aperture.

Example: What is the electric-field
pattern
of a uniformly illuminated
one-dimensional aperture of width $D$ at wavelength $\lambda$? Uniform
illumination means that the strength of the illumination is
independent of position across
the aperture:
$$g(u) = {\rm ~constant}, \qquad {-D \over 2
\lambda} < u < {+D \over 2 \lambda}$$
We will solve this problem in two
steps, first finding the far-field pattern of
a unit aperture ($D = \lambda$) and then using the similarity
theorem (below) for Fourier transforms to scale the first result to our
particular aperture.

First we calculate the Fourier
transform of the unit
rectangle
function:
$$ \Pi (u) \equiv 1\,, \qquad -1/2 < u < +1/2~,$$ $$
\Pi (u) \equiv 0 {\rm ~otherwise}\qquad\qquad\qquad$$ Notice that the
function
symbol (upper-case pi) is similar in shape to the function graph,
making it easy to
remember.

The symbol
$\Pi$ is shaped like the unit rectangle function it represents.

The function
${\rm sinc}(l) \equiv \sin(\pi l) / (\pi l)$ is the Fourier transform
of the
unit rectangle function and is the electric-field pattern of a
uniformly
illuminated unit aperture.

Next we use the powerful similarity
theorem (proof)
for Fourier transforms:

If $f(l)$ is the
Fourier transform of $g(u)$, then
$${ 1 \over
\vert a \vert} f\biggl({l \over a}\biggr)$$ is the Fourier transform of
$g(au)$, where $a \neq 0$ is a constant.

The similarity
theorem states that making a function $g$ wider or narrower makes its
Fourier transform $f$ narrower and taller or wider and shorter,
respectively,
always conserving the area under the transform. For our application, it
implies that the beamwidth
of an aperture antenna is inversely
proportional to the aperture size in wavelengths and the voltage gain
is directly proportional to the aperture size in wavelengths.

The power
pattern of a uniformly illuminated unit ($D / \lambda = 1$)
aperture. For large ($D \gg \lambda$) apertures, the zeros at $ l
= \pm 1, \pm 2, \dots$ appear at the angles $\theta = \pm \lambda / D,
\pm 2 \lambda / D, \dots$.The central peak of the power
pattern between the first minima is called the main
beam. The smaller
secondary peaks are
called sidelobes.

The weak reciprocity theorem says that
this analysis of the
transmitting power pattern also yields the receiving power pattern, or
the variation of $A_{\rm e}$ with orientation, of
an aperture antenna. In receiving terms, the power pattern represents
the point-source
response. For a uniformly illuminated aperture,
scanning a radio telescope beam in angle $\theta$ across a point source
will cause the
antenna temperature to vary as sinc$^2(\theta)$, and the half-power
response
width will equal the transmitting HPBW. The receiving HPBW is sometimes
called the resolving
power of a telescope because two equal point
sources separated by the HPBW can just be resolved by the Rayleigh
criterion that the peak response to one source coincides with
the first minimum response to the other, so the total response
has a slight minimum midway between
the
point sources.